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$y = c\cdot \sin x$, with $c = 1.7$ then no normal to the graph will also intersect the graph, but $c = 1.8$ allows this (only for a select few of the normals). What is the exact value (I'm not sure if it can be found, but I assume so) of the constant $c$ that will allow normals to also tangent the graph? In other words, what value of $c$ will allow some normals to touch the graph exactly twice(one intersection and one tangent)?

By equating $c\cdot \sin x$ with it's normal(with a parameter of $a$ for the $x$-value of the normal) you get this:

$$\sin x \cos a \cdot c^2 + x = \sin a \cos a \cdot c^2 + a$$

. For some reason I graphed this with $a$ being the $y$ axis and found that the graph looks exactly like $y = x$, until the constant $c$ is raised above $1.7$ again, then it gets really weird . Not sure if that provides anything useful but it seems important to me.

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Did you mean

$$y=c^{\sin x}$$

?

If not, wouldn't the graph be the same shape as a $\sin$ curve but stretched up and down (not sure if that was clear)?

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    Yes it is just stretching the graph from the x-axis, but no normals can intersect the graph again until c is approximately 1.7, the links go to desmos graphs that should clear that up – 2017-02-05